Analytic torsion and Ruelle zeta functions for hyperbolic manifolds with cusps

Abstract

In this paper we derive a relationship of the leading coefficient of the Laurent expansion of the Ruelle zeta function at s = 0 and the analytic torsion for hyperbolic manifolds with cusps. Here, the analytic torsion is defined by a certain regularized trace following Melrose [R.B. Melrose, The Atiyah–Patodi–Singer Index Theorem, Res. Notes Math., vol. 4, A.K. Peters, Ltd., Wellesley, MA, 1993]. This extends the result of Fried, which was proved for the compact case in [D. Fried, Analytic torsion and closed geodesics on hyperbolic manifolds, Invent. Math. 84 (3) (1986) 523–540], to a noncompact case.